There are a number of ways in the literature to find Cayley graphs with small \(\lambda (\mbox{Cay}(G,S))\) .
There are even various methods to give different sets of generators for the same group (see [1]}, [2]}).
The difficulty is to ensure that condition (REF ) is satisfied.
We will show two (actually three) ways to do so.
In all of our constructions, the elements in the sets \(B_i\)  will be of order 2, while all the elements in \(A_i\)  will be of order greater then 2.
This ensures that (REF ) is automatically satisfied.
